Use lately when you want to say something has been going on repeatedly. It's typically used with perfect tense, especially present perfect continuous. It isn't used for single events that happened recently.

correct- I've been cooking a lot lately.

correct- I've had a lot of headaches lately.

incorrect- My wife gave birth lately

Recently can replace lately, but is more commonly used for single events in the simple past.

correct- I've been cooking a lot recently. (less common than lately)

correct- Recently I've had a lot of headaches. (less common than lately)

correct- My wife gave birth recently.